On the Filtered Spectral Abscissa for Delay-Difference Equations and Its Role in the Boundary Control of Hyperbolic PDEs

Abstract.

Feedback control systems governed by delay-differential equations of neutral type may be fragile, in the sense that arbitrarily small parametric perturbations and implementation errors of the control may destroy the exponential stability of the target closed-loop system. Such instability problems can be resolved by including a low-pass filter in the control loop, on the condition that the filter itself is not destabilizing. The analysis of this condition for delay-difference equations has recently led to the notion of the filtered spectral abscissa. In this paper, we first analyze the filtered spectral abscissa, where we remove the stringent condition of commensurate delays previously made, and we derive novel mathematical and computationally tractable characterizations. Second, we highlight the role of the filtered spectral abscissa in the context of boundary control of first-order hyperbolic partial differential equations, grounded in integral transformations that result in delay-difference equation models with both discrete and distributed delays. In particular, we show that in the situation where reflection terms cannot be robustly canceled out by the control—that is, their direct cancellation would lead to a fragile closed-loop system—a negative filtered spectral abscissa of the delay-difference equation, obtained by removing the distributed delay terms, is the necessary and sufficient condition—in addition to the exponential stability of the target closed-loop system—for the safe inclusion of a filter with sufficiently high cut-off.